Question

Circle A has a center at `(2 ,8 )` and a radius of `2`. Circle B has a center at `(-8 ,3 )` and a radius of `1`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

no overlap , ≈ 8.18

Explanation:

What we have to do here is compare the distance (d ) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

To calculate d use the `color(blue)"distance formula"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 points"`

let `(x_1,y_1)=(2,8)" and " (x_2,y_2)=(-8,3)`

`d=sqrt((-8-2)^2+(3-8)^2)=sqrt(100+25)≈11.18`

radius of A + radius of B = 2 + 1 = 3

Since sum of radii < d , then no overlap

smallest distance = d - sum of radii = 11.18 - 3 = 8.18
graph{(y^2-16y+x^2-4x+64)(y^2-6y+x^2+16x+72)=0 [-20, 20, -10, 10]}